Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17405 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#21832 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×C22⋊C4 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C24×C5 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C5×#117 ◁ G | |
914 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7)(8,14)(9,12)(10,13)(11,16)(15,20)(17,19)(18,21)(22,23,25,29,26,28,32,35)(24,27,31,34,30,33,36,37);(6,8,10,16)(7,11,13,14)(9,17,18,15)(12,20,21,19)(22,24)(23,27)(25,31)(26,30)(28,33)(29,34)(32,36)(35,37);(6,9,10,18)(7,12,13,21)(8,15,16,17)(11,19,14,20)(23,28)(25,32)(27,33)(31,36);(7,13)(11,14)(12,21)(19,20)(24,30)(27,33)(31,36)(34,37)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14820);